Invariant principal order ideals under Foata's transformation (Q1953305)

Summary: Let \(\Phi\) denote Foata's second fundamental transformation on permutations. For a permutation \(\sigma\) in the symmetric group \(S_n\), let \(\widetilde{\Lambda}_{\sigma}=\{\pi\in S_n\colon\pi\leq_{w} \sigma\}\) be the principal order ideal generated by \(\sigma\) in the weak order \(\leq_{w}\). \textit{A. Björner} and \textit{M. L. Wachs} [J. Comb. Theory, Ser. A 58, No. 1, 85--114 (1991; Zbl 0742.05084)] have shown that \(\widetilde{\Lambda}_{\sigma}\) is invariant under \(\Phi\) if and only if \(\sigma\) is a 132-avoiding permutation. In this paper, we consider the invariance property of \(\Phi\) on the principal order ideals \({\Lambda}_{\sigma}=\{\pi\in S_n\colon \pi\leq \sigma\}\) with respect to the Bruhat order \(\leq\). We obtain a characterization of permutations \(\sigma\) such that \({\Lambda}_{\sigma}\) are invariant under \(\Phi\). We also consider the invariant principal order ideals with respect to the Bruhat order under Han's bijection \(H\). We find that \({\Lambda}_{\sigma}\) is invariant under the bijection \(H\) if and only if it is invariant under the transformation \(\Phi\).